QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
TEODOR BANICA AND DEBASHISH GOSWAMI
Abstract. We introduce and study two new examples of noncommutative spheres:
the half-liberated sphere, and the free sphere. Together with the usual sphere, these
two spheres have the property that the corresponding quantum isometry group is
“easy”, in the representation theory sense. We present as well some general comments
on the axiomatization problem, and on the “untwisted” and “non-easy” case.
Introduction
The aim of the present paper is to bring some contributions to the theory of noncommutative spheres, by using a number of ideas and tools coming from the recent
work on quantum isometry groups [9], [21], and on easy quantum groups [6], [7].
The noncommutative spheres were introduced by Podleś in [22], as twists of the
usual spheres. The natural framework for the study of such noncommutative objects
is Connes’ noncommutative geometry [13]. This has led to a systematic study of
the associated spectral triples, with the explicit computation of a number of related
Riemannian geometric invariants. See Connes and Dubois-Violette [14], [15], Connes
and Landi [16], Dabrowski, D’Andrea, Landi and Wagner [18].
A useful, alternative point of view comes from the relationship with the quantum
groups. The structure of the usual sphere S n−1 is intimately related to that of the
orthogonal group On , and when twisting the sphere the orthogonal group gets twisted
as well, and becomes a quantum group. See Varilly [23].
The recently developed theory of quantum isometry groups [9], [21] provides a good
abstract framework for the study of the exact relationship between noncommutative
spheres and quantum groups. Some key preliminary results in this direction were
obtained in [8], where it was shown that the usual spheres do not have quantum
symmetry, and in [10], where the case of the Podleś sphere is studied in detail.
In this paper we present some more general results in this direction, mixing spectral
triple and quantum group techniques. The idea is that the “easy” (and “untwisted”)
orthogonal quantum groups, introduced in [6] and classified in [7], are as follows: the
orthogonal group On , the half-liberated orthogonal group On∗ , and the free orthogonal group On+ . Together with the above considerations, this suggests that in the
2000 Mathematics Subject Classification. 58J42 (46L65, 81R50).
Key words and phrases. Quantum group, Noncommutative sphere.
1
2
TEODOR BANICA AND DEBASHISH GOSWAMI
“untwisted” and “easy” case we should have exactly 3 examples of noncommutative
spheres: the usual sphere S n−1 , a half-liberated sphere S∗n−1 , and a free sphere S+n−1 .
We will present here a number of results in this direction, notably by introducing and
studying in detail the half-liberated sphere S∗n−1 , and the free sphere S+n−1 .
The paper is organized as follows: in 1-2 we discuss the construction of the three
spheres, in 3-5 we study the associated projective spaces and spherical integrals, and
in 6-7 we work out the relation with the quantum isometry groups.
The final section, 8, contains a few concluding remarks.
Acknowledgements. We would like to thank J. Bichon and S. Curran for several
useful discussions. T.B. was supported by the ANR grants “Galoisint” and “Granma”,
and D.G. was supported by the project “Noncommutative Geometry and Quantum
Groups”, funded by the Indian National Science Academy.
1. The usual sphere
The simplest example of noncommutative sphere is the usual sphere S n−1 ⊂ Rn .
Our first goal will be to find a convenient functional analytic description of it. In this
section we present a series of 5 basic statements in this direction, all well-known, and
all to be extended in the next section to the noncommutative setting.
Theorem 1.1. The sphere S n−1 ⊂ Rn is the spectrum of the C ∗ -algebra
X
An = C ∗ x1 , . . . , xn xi = x∗i , xi xj = xj xi ,
x2i = 1
generated by n self-adjoint commuting variables, whose squares sum up to 1.
Proof. The first remark is that the algebra in the statement is indeed well-defined, due
to the condition Σ x2i = 1, which shows that we have ||xi || ≤ 1 for any C ∗ -norm.
Consider the algebra A0n = C(S n−1 ), with the standard coordinates denoted x0i . By
the universal property of An we have a morphism An → A0n mapping xi → x0i .
On the other hand, by the Gelfand theorem we have An = C(X) for a certain
compact space X, and we can define a map X → S n−1 by p → (x1 (p), . . . , xn (p)). By
transposing we get a morphism A0n → An mapping x0i → xi , and we are done.
The second ingredient that we will need is a functional analytic description of the
action of the orthogonal group On on S n−1 . This action can be seen as follows.
Theorem 1.2. We have a coaction Φ : An → C(On ) ⊗ An given by
X
Φ(xi ) =
uij ⊗ xj
j
where uij ∈ C(On ) are the standard matrix coordinates: uij (g) = gij .
Proof. Consider indeed the action map On × S n−1 → S n−1 , given by (g, p) → gp. By
transposing we get a coaction map Φ as in the statement.
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
3
The uniform measure on S n−1 is the unique probability measure which is invariant
under the action of On . In functional analytic terms, the result is as follows.
Theorem 1.3. There is a unique positive unital trace tr : An → C satisfying the
invariance condition (id ⊗ tr)Φ(x) = tr(x)1.
Proof. We can define indeed tr : An → C to be the integration with respect to the
uniform measure on S n−1 : the positivity condition follows from definitions, and the
invariance condition as in the statement follows from dp = d(gp), for any g ∈ On .
Conversely, it follows from the general theory of the Gelfand correspondence that a
trace as in the statement must come from the integration with respect to a probability
measure on S n−1 which is invariant under On , and this gives the uniqueness.
Theorem 1.4. The canonical trace tr : An → C is faithful.
Proof. This follows from the well-known fact that the uniform measure on the sphere
takes a nonzero value on any open set.
Finally, we have the well-known result stating that S n−1 can be identified with the
first slice of On . In functional analytic terms, the result is as follows.
Theorem 1.5. The following algebras, with generators and traces, are isomorphic:
(1) The algebra An , with generators x1 , . . . , xn , and with the trace functional.
(2) The algebra Bn ⊂ C(On ) generated by u11 , . . . , u1n , with the integration.
Proof. From the universal property of An we get a morphism π : An → Bn mapping
xi → u1i . The invariance property of the integration functional I : C(On ) → C shows
that tr0 = Iπ satisfies the invariance condition in Theorem 1.3, so we have tr = tr0 .
Finally, from the faithfulness of tr we get that π is an isomorphism, and we are done. 2. Noncommutative spheres
We are now in position of introducing two basic examples of noncommutative spheres:
the half-liberated sphere, and the free sphere. The idea will be of course to weaken or
simply remove the commutativity conditions in Theorem 1.1.
Definition 2.1. We consider the universal C ∗ -algebras
X
∗
∗
∗
2
xi = 1
An = C x1 , . . . , xn xi = xi , xi xj xk = xk xj xi ,
X
∗
A+
x1 , . . . , xn xi = x∗i ,
x2i = 1
n = C
generated by n self-adjoint variables whose squares sum up to 1, subject to the halfcommutation relations abc = bca, and to no relations at all.
4
TEODOR BANICA AND DEBASHISH GOSWAMI
Our next goal will be to find suitable analogues of Theorems 1.2, 1.3, 1.4, 1.5. For
this purpose, we will need appropriate “noncommutative versions” of the orthogonal
group On . So, let us consider the following universal algebras:
C(On∗ ) = C ∗ u11 , . . . , unn uij = u∗ij , uij ukl ust = ust ukl uij , ut = u−1
∗
+
C(On ) = C u11 , . . . , unn uij = u∗ij , ut = u−1
These algebras, introduced in [6], [24], are Hopf algebras in the sense of Woronowicz
[25], [26]. We refer to the recent paper [7] for a full discussion here.
With these definitions in hand, we can state and prove now an analogue of Theorem
1.2. We agree to use the generic notation A×
n for the 3 algebras constructed so far.
×
×
Theorem 2.2. We have a coaction Φ : A×
n → C(On ) ⊗ An given by
X
Φ(xi ) =
uij ⊗ xj
j
where uij ∈
C(On× )
are the standard generators.
Proof. We have to construct three maps, and prove that they are coactions. We will
deal with all 3 cases at the same time. Consider the following elements:
X
Xi =
uij ⊗ xj
j
These elements are self-adjoint, and their squares sum up to 1. Moreover, in the case
where both sets {xi } and {uij } consist of commuting or half-commuting elements, the
set {Xi } consists by construction of commuting or half-commuting elements.
These observations give the existence of maps Φ as in the statement. The fact that
these maps satisfy the condition (∆ ⊗ id)Φ = (Φ ⊗ id)Φ is clear from definitions. Theorem 2.3. There is a unique positive unital trace tr : A×
n → C satisfying the
condition (id ⊗ tr)Φ(x) = tr(x)1.
Proof. Consider the algebra Bn× ⊂ C(On× ) generated by the elements u11 , . . . , u1n . By
×
×
the universal property of A×
n we have a morphism π : An → Bn mapping xi → u1i , and
by composing with the restriction of the Haar functional I : C(On× ) → C, we obtain a
trace satisfying the invariance condition in the statement.
As for the uniqueness part, this is a quite subtle statement, which will ultimately
come from a certain “easiness” property of our 3 spheres. Our first claim is that we
have the following key formula, where tr is the trace that we have just constructed:
(I ⊗ id)Φ = tr(.)1
In order to prove this claim, we use the orthogonal Weingarten formula [4], [2], [6].
To any integer k let us associate the following sets:
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
5
(1) Dk : all pairings of {1, . . . , k}.
(2) Dk∗ : all pairings with an even number of crossings of {1, . . . , k}.
(3) Dk+ : all noncrossing pairings of {1, . . . , k}.
The Weingarten formula tells us that the Haar integration over On× is given by the
loops(p∨q)
following sum, where Wkn = G−1
, and where the δ symbols
kn , with Gkn (p, q) = n
are 1 if all the strings join pairs of equal indices, and 0 if not:
X
I(ui1 j1 . . . uik jk ) =
δp (i)δq (j)Wkn (p, q)
p,q∈Dk×
So, let us go back now to our claim. By linearity it is enough to check the equality
on a product of basic generators xi1 . . . xik . The left term is as follows:
X
(I ⊗ id)Φ(xi1 . . . xik ) =
I(ui1 j1 . . . uik jk )xj1 . . . xjk
j1 ...jk
=
X X
δp (i)δq (j)Wkn (p, q)xj1 . . . xjk
j1 ...jk p,q∈D×
k
=
X
δp (i)Wkn (p, q)
p,q∈Dk×
X
δq (j)xj1 . . . xjk
j1 ...jk
Let us look now at the last sum on the right. In the free case we have to sum
quantities of type xj1 . . . xjk , over all choices of multi-indices j which fit into our given
noncrossing pairing q, and just by using the condition Σx2i = 1, we conclude that the
sum is 1. The same happens in the classical case, with the changes that our pairing
q can now be crossing, but we can use now the commutation relations xi xj = xj xi .
Finally, the same happens as well in the half-liberated case, because the fact that our
pairing q has now an even number of crossings allows us to use the half-commutation
relations xi xj xk = xk xj xi , in order to conclude that the sum to be computed is 1.
Summarizing, in all cases the sum on the right is 1, so we get:
X
(I ⊗ id)Φ(xi1 . . . xik ) =
δp (i)Wkn (p, q)1
p,q∈Dk×
On the other hand, another application of the Weingarten formula gives:
tr(xi1 . . . xik )1 = I(u1i1 . . . u1ik )1
X
=
δp (1)δq (i)Wkn (p, q)1
p,q∈Dk×
=
X
p,q∈Dk×
δq (i)Wkn (p, q)1
6
TEODOR BANICA AND DEBASHISH GOSWAMI
Since the Weingarten function is symmetric in p, q, this finishes the proof of our
claim. So, let us get back now to the original question. Let τ : A×
n → C be a trace
satisfying the invariance condition in the statement. We have:
τ (I ⊗ id)Φ(x) =
=
=
=
(I ⊗ τ )Φ(x)
I(id ⊗ τ )Φ(x)
I(τ (x)1)
τ (x)
On the other hand, according to our above claim, we have as well:
τ (I ⊗ id)Φ(x) = τ (tr(x)1)
= tr(x)
Thus we get τ = tr, which finishes the proof.
We do not have an analogue of Theorem 1.4, and best is to proceed as follows.
Definition 2.4. We agree to replace from now on A×
n with its GNS completion with
×
respect to the canonical trace tr : An → C.
We actually believe that the canonical trace is faithful on the algebraic part, so that
our replacement is basically not needed, but we don’t have a proof for this fact.
Theorem 2.5. The following algebras, with generators and traces, are isomorphic:
(1) The algebra A×
n , with generators x1 , . . . , xn , and with the trace functional.
(2) The algebra Bn× ⊂ C(On× ) generated by u11 , . . . , u1n , with the integration.
×
Proof. Consider the map π : A×
n → Bn , already constructed in the proof of Theorem
2.3. The invariance property of the integration functional I : C(On× ) → C shows that
tr0 = Iπ satisfies the invariance condition in Theorem 2.3, so we have tr = tr0 . Together
with the positivity of tr and with the basic properties of the GNS construction, this
shows that π is an isomorphism, and we are done.
3. Projective spaces
In this section we study the projective spaces associated to our noncommutative
spheres. Let us first recall that the projective space over a field F is by definition
Fn − {0}/ ∼, where x ∼ y when y = λx for some λ ∈ F. We will use the notation P n−1
for the real projective space, and Pcn−1 for the complex projective space.
Let us introduce the following definition.
Definition 3.1. We denote by Cn× the subalgebra < xi xj >⊂ A×
n , taken together with
the restriction of the canonical trace.
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
7
The noncommutative projective space that we are interested in is by definition the
spectrum of Cn× , viewed as a noncommutative compact measured space.
As a first remark, in the classical case we get indeed the real projective space.
Theorem 3.2. We have Cn = C(P n−1 ).
Proof. First, since each product of coordinates xi xj : S n−1 → R takes equal values on
p and −p, this product can be regarded as being a function on P n−1 . Now since the
collection of functions {xi xj } separates the points of P n−1 , we get the result.
Quite surprisingly, in the half-liberated case we get the complex projective space.
Theorem 3.3. We have Cn∗ = C(Pcn−1 ).
Proof. First, the half-commutation relations abc = cba give abcd = cbad = cdab for
any a, b, c, d ∈ {x1 , . . . , xn }, so the elements xi xj commute indeed with each other. We
have to prove that the Gelfand spectrum of Cn∗ =< xi xj > is isomorphic to Pcn−1 .
For this purpose, we use the isomorphism P On∗ ' P Un established in [7]. This
∗
, where u, v denote respectively the fundamental
isomorphism is given by uij ukl → vij vkl
∗
corepresentations of On , Un . By restricting attention to the first row of coordinates,
∗
this gives an embedding Cn∗ ⊂ C(P Un ), mapping xi xj → v1i v1j
.
n
n−1
Consider now the complex sphere Sc ⊂ C , with coordinates denoted z1 , . . . , zn .
By performing the standard identification v1i = zi , coming from the unitary version of
Theorem 1.5, we obtain an embedding Cn∗ ⊂ C(Scn−1 ), mapping xi xj → zi z̄j .
The image of this embedding is the subalgebra of C(Scn−1 ) generated by the functions
zi z̄j . By using the same argument as in the real case, this gives the result.
In view of the above results, it is tempting to conjecture that the “threefold way” that
we are currently developing for quantum groups, noncommutative spheres and noncommutative projective spaces is actually part of the usual real/complex/quaternionic
“threefold way”, originally discovered by Frobenius, and known to play a fundamental
role in mathematical physics, according to Dyson’s paper [20].
We have here the following question.
Question 3.4. Do we have Cn+ = C(Pkn−1 )?
This is of course a quite vague question. The symbol Pkn−1 on the right is supposed
to correspond to some kind of tricky “quaternionic projective space”. Note that the
space Pkn−1 = Kn − {0}/ ∼ appearing in the existing literature won’t be suitable for
our purposes, simply because the algebra Cn+ on the left is noncommutative. What we
would need is rather a “twist”, in the spirit of the quantum projective spaces in [17].
The main problem here is to construct a representation of Cn+ , by using the Pauli
matrices. With a bit of luck, this representation can be shown to be faithful, and the
corresponding result can be interpreted as answering the above question.
8
TEODOR BANICA AND DEBASHISH GOSWAMI
A first piece of evidence comes from [3], where a certain faithful representation of
C(S4+ ) is constructed, by using the Pauli matrices. This is probably quite different
from what we need, but the main technical fact, namely that “the combinatorics of the
noncrossing partitions can be implemented by the Pauli matrices”, is already there.
A second piece of evidence comes from the results in [7], which suggest that the free
quantum groups might be actually supergroups. Once again, this kind of argument is
quite speculative, and maybe a bit far away from the present considerations.
Let us end this section by recording a few modest facts about Cn+ .
Proposition 3.5. The algebras Cn+ are as follows:
(1) At n = 2 we have C2+ = C2∗ .
(2) At n ≥ 3 we have Cn+ 6= Cn∗ .
Proof. (1) This follows either from the isomorphism O2+ = O2∗ established in [7], or
directly from definitions, by using the fact that C2+ is commutative.
(2) It is enough here to prove that C3+ is not commutative. For this purpose, we will
use the positive matrices in M2 (C). These are matrices of the following form:
p a
Y =
ā q
Here p, q ∈ R and a ∈ C must be chosen such that both eigenvalues are positive,
and this happens for instance when p, q > 0 and a ∈ C is small enough.
Let us fix some numbers pi , qi > 0 for i = 1, 2, 3, satisfying Σpi = Σqi = 1. For any
choice of small complex numbers ai ∈ C satisfying Σai = 0, the corresponding elements
Yi constructed as above will be positive, and will sum up to 1. Moreover, by carefully
choosing the ai ’s, we can arrange as √
for Y1 , Y2 , Y3 not to pairwise commute.
Consider now the matrices Xi = Yi . These are all self-adjoint, and their squares
sum up to 1, so we get a representation A+
3 → M2 (C) mapping xi → Xi . Now this
representation restricts to a representation C3+ → M2 (C) mapping x2i → Yi , and since
the Yi ’s don’t commute, it follows that C3+ is not commutative, and we are done. The above result suggests the following extra question regarding Cn+ : what is the
Gelfand spectrum of the algebra Cn+ /I, where I ⊂ Cn+ is the commutator ideal?
Observe that the canonical arrow Cn+ → Cn∗ and Theorem 3.3 tell us that this
Gelfand spectrum must contain Pcn−1 . Moreover, Proposition 3.5 shows that at n = 2
this inclusion is an equality. However, at n = 3 already the answer is not clear.
4. Probabilistic aspects
We know from the previous sections that we have three basic examples of “noncommutative spheres”, namely those corresponding to the algebras An , A∗n , A+
n . In this
section and in the next one we investigate the key problem of computing the integral
over these noncommutative spheres of polynomial quantities of type xi1 . . . xik .
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
9
Definition 4.1. The polynomial spherical integrals will be denoted
Z
xi1 . . . xik dx
I=
n−1
S×
with this quantity standing for the complex number obtained as image of the well-defined
×
element xi1 . . . xik ∈ A×
n by the well-defined trace functional tr : An → C.
The problem of computing such integrals has been heavily investigated in the last
years, and a number of results are available from [2], [5], [12], [19]. In what follows we
will make a brief presentation of this material, by focusing of course to the applications
to S×n−1 . We will present as well some new results, in the half-liberated case.
Let us begin our study with an elementary result.
Proposition 4.2. We have the formula
Z
xi1 . . . xik dx = 0
n−1
S×
unless each xi appears an even number of times.
Proof. This follows from the fact that for any i we have an automorphism of A×
n given
by xi → −xi . Indeed, this automorphism must preserve the trace, so if xi appears an
odd number of times, the integral in the statement satisfies I = −I, so I = 0.
The basic tool for computing spherical integrals is the Weingarten formula. Let us
recall from section 2 that associated to any integer k are the following sets:
(1) Dk : all pairings of {1, . . . , k}.
(2) Dk∗ : all pairings with an even number of crossings of {1, . . . , k}.
(3) Dk+ : all noncrossing pairings of {1, . . . , k}.
These sets can be regarded as being associated to our spheres S×n−1 , because they
come from the representation theory of the associated quantum groups On× .
Theorem 4.3. We have the Weingarten formula
Z
X
xi1 . . . xik dx =
δp (i)Wkn (p, q)
n−1
S×
p,q∈Dk×
loops(p∨q)
where Wkn = G−1
, and where the δ symbol is 1 if all the
kn , with Gkn (p, q) = n
strings of p join pairs of equal indices of i = (i1 , . . . , ik ), and is 0 if not.
Proof. This follows from the Weingarten formula in [12], [2], [4], via the identification
in Theorem 2.5, and from the fact that the Weingarten matrix is symmetric in p, q. As a first application, we have the following result.
Theorem 4.4. With n → ∞, the standard coordinates of S×n−1 are as follows:
10
TEODOR BANICA AND DEBASHISH GOSWAMI
(1) Classical case: real Gaussian, independent.
(2) Half-liberated case: symmetrized Rayleigh, their squares being independent.
(3) Free case: semicircular, free.
Proof. This follows from Theorem 4.3 and from the fact that Wkn is asymptotically
diagonal, see [2], [6]. The only new assertion is the independence one in (2), which can
be proved as in [6], by using the fact that the mixed cumulants vanish.
Note that the independence in (1,2) follows as well from the exact formulae in Theorems 5.1 and 5.2 below, by letting n → ∞ and by using the Stirling formula.
5. Spherical integrals
We discuss in this section a quite subtle problem, of theoretical physics flavor, namely
the exact computation of the polynomial integrals over S×n−1 .
In the classical case, we have the following well-known result.
Theorem 5.1. The spherical integral of xi1 . . . xik vanishes, unless each a ∈ {1, . . . , n}
appears an even number of times in the sequence i1 , . . . , ik . If la denotes this number
of occurrences, then
Z
(n − 1)!!l1 !! . . . ln !!
xi1 . . . xik dx =
(n + Σli − 1)!!
S n−1
with the notation m!! = (m − 1)(m − 1)(m − 5) . . .
Proof. The first assertion follows from Proposition 4.2. The second assertion is wellknown, and can be proved by using spherical coordinates, the Fubini theorem, and
some standard partial integration tricks. See e.g. [4].
In the case of the half-liberated sphere, we have the following result.
Theorem 5.2. The half-liberated spherical integral of xi1 . . . xik vanishes, unless each
number a ∈ {1, . . . , n} appears the same number of times at odd and at even positions
in the sequence i1 , . . . , ik . If la denotes this number of occurrences, then:
Z
(2n − 1)!l1 ! . . . ln !
xi1 . . . xik dx = 4Σli
(2n + Σli − 1)!
S∗n−1
Proof. First, by using Proposition 4.2 we see that the integral I in the statement
vanishes, unless k = 2l is even. So, assume that we are in the non-vanishing case. By
using Theorem 3.3 the corresponding integral over the complex projective space Pcn−1
can be viewed as an integral over the complex sphere Scn−1 , as follows:
Z
I=
zi1 z̄i2 . . . zi2l−1 z̄i2l dz
Scn−1
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
11
Now by using the same argument as in the proof of Proposition 4.2, but this time
with transformations of type p → λp with |λ| = 1, we see that I vanishes, unless each
za appears as many times as z̄a does, and this gives the first assertion.
Assume now that we are in the non-vanishing case. Then the la copies of za and the
la copies of z̄a produce by multiplication a factor |za |2la , so we have:
Z
I=
|z1 |2l1 . . . |zn |2ln dz
Scn−1
Now by using the standard identification Scn−1 ' S 2n−1 , we get:
Z
(x21 + y12 )l1 . . . (x2n + yn2 )ln d(x, y)
I =
2n−1
S
X l l Z
1
... n
x2l1 −2r1 y12r1 . . . xn2ln −2rn yn2rn d(x, y)
=
r1
rn S 2n−1 1
r1 ...rn
By using the formula in Theorem 5.1, we get:
X l1 ln (2n − 1)!!(2r1 )!! . . . (2rn )!!(2l1 − 2r1 )!! . . . (2ln − 2rn )!!
I =
...
r1
rn
(2n + 2Σli − 1)!!
r1 ...rn
X l1
(2n − 1)!(2r1 )! . . . (2rn )!(2l1 − 2r1 )! . . . (2ln − 2rn )!
l
=
... n
r1
rn
(2n + Σli − 1)!r1 ! . . . rn !(l1 − r1 )! . . . (ln − rn )!
r ...r
1
n
We can rewrite the sum on the right in the following way:
X l1 ! . . . ln !(2n − 1)!(2r1 )! . . . (2rn )!(2l1 − 2r1 )! . . . (2ln − 2rn )!
I =
(2n + Σli − 1)!(r1 ! . . . rn !(l1 − r1 )! . . . (ln − rn )!)2
r1 ...rn
X 2r1 2l1 − 2r1 X 2rn 2ln − 2rn (2n − 1)!l1 ! . . . ln !
=
...
r1
l1 − r1
rn
ln − rn
(2n + Σli − 1)!
r
r
n
1
l1
ln
The sums on the right being 4 , . . . , 4 , we get the formula in the statement.
In the case of the free sphere, we already know from Theorem 4.4 that the standard coordinates x1 , . . . , xn are asymptotically semicircular and free. However, the
computation of their joint law for a fixed value of n is a well-known open problem, of
remarkable difficulty. The point is that the Gram matrix Gkn , which is nothing but Di
Francesco’s “meander matrix” in [19], cannot be diagonalized explicitely.
The best result in this direction that is known so far is as follows.
Theorem 5.3. The moments of the free hyperspherical law are given by
Z
l+1
X
1
q+1
1
r
2l + 2
r
2l
·
·
(−1)
x1 dx =
l
l + r + 1 1 + qr
n−1
(n + 1) q − 1 l + 1 r=−l−1
S+
where q ∈ [−1, 0) is given by q + q −1 = −n.
12
TEODOR BANICA AND DEBASHISH GOSWAMI
Proof. This is proved in [5], the idea being that x1 ∈ A+
n can be modelled by a certain
variable over SU2q , which can be studied by using advanced calculus methods.
Our question is whether Theorem 5.1 and Theorem 5.2 have a free analogue.
Question 5.4. Does the liberated spherical integral
Z
xi1 . . . xik dx
n−1
S+
appear as a “free analogue” of the quantities computed in Theorems 5.1 and 5.2?
The answer here is very unclear, even in the case where the indices i1 , . . . , ik are all
equal. In fact, the above question is probably closely related to Question 3.4.
Let us also mention that the meander determinant computed by Di Francesco in
[19], which appears as denominator of the abstract Weingarten-theoretical fraction
expressing the integral in Question 5.4, is a product of Chebycheff polynomials. Our
question is whether some “magic” simplification appears when computing the fraction.
In fact, our Questions 3.4 and 5.4 should be regarded as a slight, very speculative
advance on the conceptual understanding of the various formulae in [5], [19].
6. Spectral triples
In the reminder of this paper, our goal will be to study the “differential structure” of
the noncommutative spheres S×n−1 . Besides of being of independent theoretical interest,
this study will lead via the results in [7], [9] to a “global look” to our 3 spheres.
The natural framework for the study of noncommutative objects like S×n−1 is Connes’
noncommutative geometry [13], where the basic definition is as follows.
Definition 6.1. A compact spectral triple (A, H, D) consists of the following:
(1) A is a unital C ∗ -algebra.
(2) H is a Hilbert space, on which A acts.
(3) D is a (typically unbounded) self-adjoint operator on H, with compact resolvents, such that [D, a] has a bounded extension, for any a in a dense ∗-subalgebra
(say A) of A.
This definition is of course over-simplified, as to best fit with the purposes of the
present paper. We refer to [13] for the exact formulation of the axioms.
In what follows we will be mainly interested in the sphere S n−1 , and in its noncommutative versions S∗n−1 and S+n−1 . These objects are all quite simple, geometrically
speaking, and we will make only a moderate use of the general machinery in [13].
Our guiding examples, all very basic, will be as follows.
Proposition 6.2. Associated to a compact Riemannian manifold M are the following
spectral triples (A, H, D), with A and A being the algebra of continuous functions and
that of smooth functions on M respectively:
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
13
(1) H is the space of square-integrable spinors, and D is the Dirac operator.
(2) H is the space of forms on M , and D is the Hodge-Dirac operator
d + d∗ .
√
(3) H = L2 (M, dv), dv being the Riemannian volume, and D = d∗ d.
Here in the first example M is of course assumed to be a spin manifold. The fact
that all the above triples satisfy Connes’ axioms in Definition 6.1 comes from certain
standard results in global differential geometry, and we refer here to [13] and references
therein. Let us also remark that the third example, though rather uninteresting from
the viewpoint of algebraic topology or K-theory, contains all the useful information
about the Riemannian geometry of the manifold, like the volume or the curvature.
Let us go back now to our 3 noncommutative spheres, described by the algebras A×
n
in the previous sections. It is technically convenient at this point to slightly enlarge
our formalism, by starting with the following “minimal” set of axioms.
Definition 6.3. A spherical algebra is a C ∗ -algebra A, given with a family of generators
x1 , . . . , xn and with a faithful positive unital trace tr : A → C, such that:
(1) x1 , . . . , xn are self-adjoint.
(2) x21 + . . . + x2n = 1.
(3) tr(xi ) = 0, for any i.
As a first observation, each A×
n is indeed a spherical algebra in the above sense.
We know that for An = C(S n−1 ), there are at least 3 spectral triples that can be
constructed, namely those in Proposition 6.2. In the case of A∗n , A+
n , however, or more
generally in the case of an arbitrary spherical algebra, the situation with the first two
constructions is quite unclear, and the third construction will be our model.
We agree to view the identity 1 as a length 0 word in the generators x1 , . . . , xn .
Theorem 6.4. Associated to any spherical algebra A =< x1 , . . . , xn > is the compact
spectral triple (A, H, D), where the dense subalgebra A is the linear span of all the finite
words in the generators xi , and D acting on H = L2 (A, tr) is defined as follows:
(1) Let Hk = span(xi1 . . . xir |i1 , . . . , ir ∈ {1, . . . , n}, r ≤ k).
⊥
(2) Let Ek = Hk ∩ Hk−1
, so that H = ⊕∞
k=0 Ek .
(3) We set Dx = kx, for any x ∈ Ek .
Proof. We have to show that [D, Ti ] is bounded, where Ti is the left multiplication
by xi . Since xi ∈ A is self-adjoint, so is the corresponding operator Ti . Now since
⊥
Ti (Hk ) ⊂ Hk+1 , by self-adjointness we get Ti (Hk⊥ ) ⊂ Hk−1
. Thus we have:
Ti (Ek ) ⊂ Ek−1 ⊕ Ek ⊕ Ek+1
This gives a decomposition of type Ti = Ti−1 + Ti0 + Ti1 . It is routine to check that
we have [D, Tiα ] = αTiα for any α ∈ {−1, 0, 1}, and this gives the result.
As a first example, in the classical case the situation is as follows.
14
TEODOR BANICA AND DEBASHISH GOSWAMI
Theorem 6.5. For the algebra An = C(S n−1 ), the spectral triple constructed in Theorem 6.4 essentially coincides with the one described in Proposition
6.2 (3). More
√
∗
precisely, the Dirac operator
D of Theorem 6.4 is related
p to d d by the bijective cor√
1
n
∗
respondence: D = f ( d d), where f (s) = 1 − 2 + 2 4s2 + (n − 2)2 , s ∈ [0, ∞). In
√
particular, the eigenspaces of D and d∗ d coincide.
√
Proof. This follows from the well-known fact that d∗ d diagonalizes as in Theorem
6.4, with the corresponding eigenvalues being k(k + n − 2), with k = 0, 1, 2, . . .
7. Quantum isometries
We know from the previous section that associated to any spherical algebra A, and
in particular to the algebras A×
n , is a certain spectral triple (A, H, D). In the
√ classical
case A = An this spectral triple is the one coming from the operator D = d∗ d.
Let us recall now the definition of the quantum isometry groups from [9], slightly
modified as to fit with our setting. Let S = (A, H, D) be a spectral triple of compact
type, with H assumed to be the GNS space of a certain faithful trace tr : A → C.
Consider the category of compact quantum groups acting on S isometrically, that
is, the compact quantum group (say Q) must have a unitary representation U on H
which commutes with D, satisfies U 1A = 1Q ⊗ 1A and adU maps A00 into itself.
If this category has a universal object, then this universal object (which is unique
up to isomorphism) will be denoted by QISO(S). See [9] for more details.
Proposition 7.1. Let A be a spherical algebra, and consider the associated spectral
triple S = (A, H, D). Then QISO(S) exists.
Proof. The proposition follows from Theorem 2.24 of [9], since the linear space spanned
by 1A is an eigenspace of D.
Theorem 7.2. QISO(S×n−1 ) = On× .
×
×
Proof. Consider the standard coaction Φ : A×
n → C(On ) ⊗ An . This extends to a
×
unitary representation on the GNS space Hn , that we denote by U .
We have Φ(Hk ) ⊂ C(On× ) ⊗ Hk , which reads U (Hk ) ⊂ Hk . By unitarity we get as
well U (Hk⊥ ) ⊂ Hk⊥ , so each Ek is U -invariant, and U, D must commute. That is, Φ is
isometric with respect to D, and On× must be a quantum subgroup of QISO(S×n−1 ).
Assume now that Q is compact quantum group with a unitary representation V
00
on H × commuting with D, such that adV leaves (A×
n ) invariant. Since D has an
∗
eigenspace consisting exactly of x1 , . . . , xn , both V and V must preserve this subspace,
so we can find self-adjoint elements bij ∈ C(Q) such that:
X
adV (xi ) =
bij ⊗ xj .
j
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
15
From the unitarity of V , it is also easy to see that adV is trace-preserving, and by
using this it follows that ((bij )) as well as ((bji )) are unitaries. It follows in particular
that the antipode κ of Q must send bij to bji . Moreover, using the defining relations
satisfied by the xi ’s and the fact that adV and (κ ⊗ id) ◦ adV are ∗-homomorphism,
we can prove that the bij ’s will satisfy the same relations as those of the generators
uij of C(On× ). Indeed, for the free case there is nothing to prove, and we have verifed
such relations for the classical (commutative) case, i.e. for C(S n−1 ), in [8], the proof
of which will go through almost verbatim for the half-liberated case too, replacing the
words xi xj of length two by the length-3 words xi xj xk . This shows that C(Q) is a
quotient of C(On× ), so Q is a quantum subgroup of On× , and we are done.
There are several questions raised by the above results, concerning the axiomatization
of the noncommutative spheres. Perhaps the most important is the following one:
Question 7.3. What conditions on a spherical algebra A ensure the fact that the
corresponding quantum isometry group is “easy” in the sense of [6]?
An answer here would of course provide an axiomatization of the “easy spheres”,
and our above results would translate into a 3-fold classification for the easy spheres,
because of the classification results for easy quantum groups in [7].
8. Concluding remarks
We have seen in this paper that the usual sphere S n−1 , the half-liberated sphere
S∗n−1 , and the free sphere S+n−1 , share a number of remarkable common properties.
The general axiomatization and study of these 3 noncommutative spheres has raised
a number of concrete questions, notably in connection with the general structure of
the associated projective spaces (Question 3.4), with the computation of the associated spherical integrals (Question 5.4), and with the general axiomatization problem
(Question 7.3). We intend to come back to these questions in some future work.
In addition, there are many questions about what happens in the “untwisted” case,
and in the “non-easy” case. Some results here are already available from [10].
Finally, we have the more general problem of understanding the notion of liberation
and half-liberation for more general manifolds. In the 0-dimensional case it is probably
possible to use the results in [11] in order to reach to some preliminary results. In the
continuous case, however, the situation so far appears to be quite unclear.
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T.B.: Department of Mathematics, Toulouse 3 University, 118 route de Narbonne,
31062 Toulouse, France. [email protected]
D.G.: Theoretical Statistics and Mathematics Unit, 203 Barrackpore Trunk Road,
Kolkata 700 108, India. [email protected]